In a Catholic Mass, why does only the priest receive Communion under both kinds (under the species of bread and wine)?
Communion is always "valid under both species" as Catholics would say. That means that if you only receive Jesus under the form of wine or under the form of bread, you're receiving the whole of God.
For about 20 years (depending upon where you live) in much of North America it has been common practice to offer the cup to the laity during communion for all masses on every day of the week. This is extra-ordinary, and not necessary and seems to be going away. The diocese of Phoenix was the first to go back to the old norms and my diocese (Madison, WI) just went back starting this past Advent.
The rationale our priest gave was that the bishop felt that absence would make the heart grow fonder - on those occasions when we could receive from the cup we'd be more apt to do so. When half of the congregation just passes the Eucharistic minister by like there's nothing in the cup, you can see why absence may be necessary for a time. Hopefully, come Holy Thursday or the feast of Corpus Christi we can get a few more people considering Who they're passing up.
The other good (well, I think it's good) thing the new reversion in norms does is limit the excruciating number of Eucharistic ministers (I mean extra ordinary ministers of Holy Communion). Any time you have to have a dispensation to receive an indult you're looking at an institution within the church that won't last. Nothing doctrinal about it, the Church is free to let lay people distribute communion and receive in both species as well as end that practice.
If there were enough priests and deacons to distribute communion in the first place the practice - distributing both the body and the blood at daily mass as well - might have continued or even spread, I'd imagine.
Why does only the priest receive Communion under both kinds at a Catholic Mass?
The faithful may receive communion under both species (bread and wine) at mass. But a priest must consume both species at mass.
The dogmatic teaching of the Church
Holy Communion must of necessity be received by all adults in virtue of the divine command: "You can have no life in yourselves, unless you eat the flesh of the Son of Man, and drink his blood" (John vi, 54). There is also the precept of the Church (Denz. 437, 1205 sq).
Furthermore, St. Thomas maintains (Summa, Ilia, q. 73, a. 3) that Holy Communion is necessary for salvation independently of the divine command, for the Eucharist is in itself an essential means of sanctification, "of completing the spiritual life, and thus it is the end of all the other sacraments" in so far as of its very nature it increases charity in our souls and unites us to Christ the Saviour. "You can have no life in yourselves, unless you eat the flesh of the Son of Man, and drink his blood" (John vi, 54). Therefore, according to St. Thomas, all the faithful without exception must receive the effect of Holy Communion by an implicit desire at least. The grace of Baptism is ordered to this effect in the same way as a child tends towards manhood. But if this is true for all the faithful in general, with even greater reason is it true for the priest.
Moreover, the celebrant's Communion is required for the completion of the sacrifice of the Mass; Christ is present under the species of the eucharistic sacrifice after the manner of food and drink, and so is meant to be consumed.
Finally, the priest must receive Communion under both species: "So far as the sacrament itself is concerned it is most fitting that both the body and the blood be received since the perfection of the sacrament rests in both, and therefore, because it is the priest's duty both to consecrate and to complete the sacrament, lie should never receive the body of Christ without the blood" (Summa, Ilia, q. 80, a. 12; cf. also ad 1). The priest has a divine obligation in this respect.
St. Thomas continues in the article just quoted: "On the part of the communicants extreme reverence and caution are necessary to prevent anything happening which would not be fitting for so great a sacrament. This is most likely to occur in the drinking of the blood which might easily be spilt, if there were any carelessness in receiving it. And because the Christian community has increased ... it is a prudent custom for the blood to be received by the priest alone and not by the people."
Ibid. ad. 3: "Nothing is lost by this (that is, by the body being received by the people without the blood): because the priest both offers and receives the blood in the name of all, and the whole Christ is present under either species." Under the species of bread there is also present, by concomitance, the precious blood. Thus the faithful are not deprived of any notable grace, and a fervent Communion under one species is far more fruitful than a tepid Communion received under both species.
It is not a matter of the priest being the only one who can drink from the chalice. The faithful normally do not receive from the chalice because it is not necessary and because many parishes are not designed in a way that permits for orderly and smooth reception of both species.
For further information about this subject matter, the following may be of interest:
OP: Why does only the priest receive Communion under both kinds at a Catholic Mass?
It is an interesting question with some reasonable explanations, yet only one answer makes the most sense historically. In fact, this one answer could have drove the other reasons.
In case some believe this issue was of no importance, we should understand that Jan Huss was burned at the stake for apparently teaching that both the bread and wine should be given to the laity, as well as the clergy. About 100 years later, this led to this part of the Augsburg Confession. Basically, this issue helped develop the Reformation.
Augsburg Confession 1530 AD Part Second Article I. Of both Kinds [in the Lord’s Supper] Both kinds of the Sacrament in the Lord's Supper are given to the laity, because that this custom hath the commandment of the Lord: 'Drink ye all of this' (Matt. xxvi. 27); where Christ doth manifestly command concerning the cup that all should drink. And that no man might cavil that this doth only pertain to the priests, the example of Paul to the Corinthians witnesseth that the whole Church did use both kinds in common (1 Cor. xi. 28). And this custom remained a long time in the Church; neither is it certain when or by what authority it was changed; although the Cardinal de Cusa relates when it was approved. [And this custom remained a long time in the churches, as may be proved from history and the writings of the Fathers.] Cyprian in certain places doth witness that the blood was given to the people; the same thing doth Jerome testify, saying, 'The priests do minister the Eucharist, and communicate the blood of Christ to the people.' Nay, Pope Gelasius commandeth that the Sacrament be not divided (Dist. II., De Consecr. Cap. Comperimus). Only a custom, not thus ancient, doth otherwise. But it is manifest that a custom, brought in contrary to the commandments of God, is not to be approved, as the Canons do witness (Dist. VIII., Cap. Veritate) with the words which follow. Now this custom has been received, not only against the Scripture, but also against the ancient Canons and the example of the Church. Therefore if any would rather use both kinds in the Sacrament, they are not to be compelled to do otherwise with the offense of their conscience. And because that the division of the Sacrament doth not agree with the institution of Christ, among us it is the custom to omit that procession which hitherto hath been in use. https://ccel.org/ccel/schaff/creeds3/creeds3.iii.ii.html
At that time and earlier, the Catholic Church was teaching otherwise, believing the custom, even dogma, of offering bread only to the laity was already binding. The reasons were for greater profit and better protection (bold mine).
according as she [Catholic Church] judges it expedient for the greater profit of the recipients or the better protections of the sacraments themselves against irreverence. Hence "although the usage of Communion under two kinds was not infrequent in the early ages [ab initio] of the Christian religion, yet, the custom in this respect having changed almost universally [latissime] in the course of time, holy mother the Church, mindful of her authority in the administration of the Sacraments, and influenced by weighty and just reasons, has approved the custom of communicating under one kind, and decreed it to have the force of a law, which may not be set aside or changed but by the Church's own authority" (Trent, Sess. XXI, c. ii). Not only, therefore, is Communion under both kinds not obligatory on the faithful, but the chalice is strictly forbidden by ecclesiastical law to any but the celebrating priest. These decrees of the Council of Trent were directed against the Reformers of the sixteenth century, who, on the strength of John 6:54, Matthew 26:27, and Luke 22:17-19, enforced in most cases by a denial of the Real Presence and of the Sacrifice of the Mass, maintained the existence of a Divine precept obliging the faithful to receive under both kinds, and denounced the Catholic practice of withholding the cup from the laity as a sacrilegious mutilation of the sacrament. A century earlier the Hussites, particularly the party of the Calixtines, had asserted the same doctrine, without denying, however, the Real Presence or the Sacrifice of the Mass, and on the strength principally of John 6:54; and the Council of Constance in its thirteenth session (1415) had already condemned their position and affirmed the binding force of the existing discipline in terms practically identical with those of Trent (see decree approved by Martin V, 1418, in Denzinger, Enchiridion, n. 585). It is to be observed that neither council introduced any new legislation on the subject; both were content with declaring that the existing custom had already acquired the force of law. https://www.newadvent.org/cathen/04175a.htm
The first reasons, greater profit and better protection, are the most common answers to the question. The real reason is about the chalice, which will be explained in more detail.
The explanations spring from the Catholic view of transubstantiation.
Transubstantiation is a three-part teaching that only a duly ordained priest acting in the stead of Christ on earth with right intention at the proper altar could bread be turned into the Body of Christ and wine turned into the Blood of Christ. The accidents of bread and wine, what you see and taste, remain, but the substance, what you don’t see or taste, has changed. They further teach that both species of Body and Blood are fully present in either confected bread or wine.
1411 Only validly ordained priests can preside at the Eucharist and consecrate the bread and the wine so that they become the Body and Blood of the Lord. CCC http://www.scborromeo.org/ccc/p2s2c1a3.htm#VI
This consecration of priests suggests the same with the consecration of vessels, fitting hand in glove.
The term transubstantiation seems to have been first used by Hildebert of Tours (about 1079). His encouraging example was soon followed by other theologians, as Stephen of Autun (d. 1139), Gaufred (1188), and Peter of Blois (d. about 1200), whereupon several ecumenical councils also adopted this significant expression, as the Fourth Council of the Lateran (1215), and the Council of Lyons (1274), in the profession of faith of the Greek Emperor Michael Palæologus. The Council of Trent (Sess. XIII, cap. iv; can. ii) [1545-1563] not only accepted as an inheritance of faith the truth contained in the idea, but authoritatively confirmed the "aptitude of the term" to express most strikingly the legitimately developed doctrinal concept. https://www.newadvent.org/cathen/05573a.htm#section3
With transubstantiation in mind, one may understand the Catholic Church’s desire for proper decorum and careful usage and worship of the Host.
When the duly ordained priest spoke certain intentional words, they believe the bread became the Body. It was then lifted up and adored. This practice took place before the invocation of the wine. This was called the Elevation of the Host, which led to exposition or display of the Bread.
There can be no reasonable doubt that the practice of exposition [of the Bread] came in the wake of that most epoch-making liturgical development, the Elevation of the Host in the Mass. The Elevation itself, of which we first hear in its present sense about the year 1200, was probably adopted as a practical protest against the teaching of Peter Comestor and Peter the Chanter, who held that the bread was not consecrated in the Mass until the words of institution had been spoken over both bread and wine. Those who believed that when the words "Hoc est enim corpus meum" had been pronounced, the bread was at once changed into the flesh of our Lord, supported their opinion by adoring the Sacrament, and holding It up for the adoration of the people, without waiting for the words to be spoken over the chalice. At Paris, this elevation became a matter of synodal precept, probably before the year 1200. Before long it came to be regarded as a very meritorious act to look upon and salute the Body of the Lord. https://www.newadvent.org/cathen/05713a.htm
Thus, one may reason that placing the bread directly on the tongue helps to avoid crumbs. To avoid spillage of the wine, it may be best not to share a chalice, although the real reason was already hinted at.
To be sure, it was necessary for the theologians to explain that the Body and Blood were fully present in each drop and crumb. It was also necessary to dismiss the historic practice of partaking in both. See again the article on transubstantiation.
In short, the laity came to receive the consecrated Bread as it was worshipped before the wine was transformed. It was explained that all of Christ was in the Bread. As such, caution had to be taken.
So, those are the reasons given from about the year 1000 on as to why the laity should receive only the bread in a certain manner, but why does the priest receive both?
We find another hint some 200 years later about this desire to avoid dangers and scandals during the Council of Constance in 1418.
Moreover, just as this custom [offering only bread to the laity] was sensibly introduced in order to avoid various dangers and scandals, so with similar or even greater reason was it possible to introduce and sensibly observe the custom that, although this sacrament was received by the faithful under both kinds in the early church, nevertheless later it was received under both kinds only by those confecting it, and by the laity only under the form of bread. https://www.papalencyclicals.net/councils/ecum16.htm
What dangers? Certainly, the caution required to worship the Bread without losing a crumb. But what scandals? The question of superiority and inferiority between priest and laity had clearly arisen by that time. But the word “confecting” tells us directly; that is, Consecration, which is the separation from the profane to the sacred. Certainly, Catholic priests are separated, consecrated as it were. It is the same with the confected bread, but also the consecrated chalice.
Before the chalice and paten are used in the Sacrifice of the Mass they require consecration. This rite is carried out according to a form specially provided in the "Pontificale" and involving the use of holy chrism. The consecration must be performed by a bishop (or in the case of chalices intended for monastic use, by an abbot possessing the privilege), and a bishop cannot in an ordinary way delegate any priest to perform this function in his place. Further, if the chalice lose its consecration — which happens for example if it be broken or the cup perforated, or even if it has had to be sent to have the bowl regilded—it is necessary that it should be reconsecrated by the bishop before it can again be used. Strictly speaking, only priests and deacons are permitted to touch the chalice or paten, but leave is usually granted to sacristans and those officially appointed to take charge of the vestments and sacred vessels. https://www.newadvent.org/cathen/03561a.htm
You may be able to clearly see the problem that has been hinted at. A consecrated chalice is only to be touched by the duly ordained and not by the, what might be, unclean laity. While the Host may be placed directly on the tongue and thus avoid any laity touch of the paten to avoid contamination, how would wine be given from a chalice not to be touched?
This consecrated idea of both the clergy and vessels dates far back in time to Pope Sixtus I circa 120 CE.
According to the "Liber Pontificalis" (ed. Duchesne, I, 128), he [Pope Sixtus I] passed the following three ordinances: (1) that none but sacred ministers are allowed to touch the sacred vessels; https://www.newadvent.org/cathen/14031b.htm
The first List of Popes was written about 530 CE. It may have been post dated to Sixtus I to give more weighty authority to the idea. Perhaps this time frame circa 600 CE is when the practice of withholding the chalice began. As the Catholic Church says, the practice was already a law long before 1200 CE.
So, to answer the OP, the primary, indeed real, reason the laity could not drink from the chalice is one of consecration. The laity are common, not consecrated. But the priests could touch the chalices because not only are the chalices consecrated, but so are the sacred ministers.